Optimal. Leaf size=64 \[ -\frac {\, _2F_1\left (\frac {1}{2},n;1+n;\sec (e+f x)\right ) (d \sec (e+f x))^n \tan (e+f x)}{f n \sqrt {1-\sec (e+f x)} \sqrt {1+\sec (e+f x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3891, 66}
\begin {gather*} -\frac {\tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left (\frac {1}{2},n;n+1;\sec (e+f x)\right )}{f n \sqrt {1-\sec (e+f x)} \sqrt {\sec (e+f x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 3891
Rubi steps
\begin {align*} \int (d \sec (e+f x))^n \sqrt {1+\sec (e+f x)} \, dx &=-\frac {(d \tan (e+f x)) \text {Subst}\left (\int \frac {(d x)^{-1+n}}{\sqrt {1-x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {1-\sec (e+f x)} \sqrt {1+\sec (e+f x)}}\\ &=-\frac {\, _2F_1\left (\frac {1}{2},n;1+n;\sec (e+f x)\right ) (d \sec (e+f x))^n \tan (e+f x)}{f n \sqrt {1-\sec (e+f x)} \sqrt {1+\sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 67, normalized size = 1.05 \begin {gather*} \frac {2 \, _2F_1\left (\frac {1}{2},1-n;\frac {3}{2};1-\sec (e+f x)\right ) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \sin (e+f x)}{f \sqrt {1+\sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (d \sec \left (f x +e \right )\right )^{n} \sqrt {1+\sec \left (f x +e \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \sec {\left (e + f x \right )}\right )^{n} \sqrt {\sec {\left (e + f x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {\frac {1}{\cos \left (e+f\,x\right )}+1}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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